1. Use the integral test to determine whether the series is convergent or divergent.

__1__ (where the summation sign is infront of it, starting at n=1 --> infinity)
n^5

2. The Rule of 70 in finance uses the approximation ln(1 + i) = i when i is small. In fact, the first degree Taylor polynomial for f(x)=ln(1 + x) centered at a = 0 is Tsubone(x) = x.
(a) Find the 3rd degree Taylor polynomial for f(x) = ln(1+x), centered at a = 0.
(b) Estimate the remainder Rsubthree(x) over the interval -.5<x<.5 using Taylors Inequality.
(c) Use your answer from part (b) to estimate the size of the error Rsubthree(0.1)
(d) With your calculator, approximate the remainder Rsubthree(0.1) = f(0.1) - Tsubthree(0.1)
(e) Compare your answersfrom parts (c) and (d)


Any Help is Appreciated, please show your steps/work so I can see how you arrived at your answer.
Thanks for your time guys

for the first, look at the p test.

i haven't looked at it in a decade, but off the top of my head,

sum x = 1 to inf [1 / x^p] converges for p > 1

first you need to show its positive decreasing and continuous otherwise the integral test does not apply

d/dx is -5/x^6 clearly it is decreasing

its continuous on your interval surely too

positive is obvious the main thing is just to show it is decreasing by taking the derivative

you can just knock the others out by a comment about it never being negative or not continuous

f(x) = 1/x^5

(integral from 1 to 00) = lim t->00 int 1 to 00 of 1/x^5 dx

lim t -> 00 -1/4t^4 - -1/4

1/4

hence the integral is convergent and by the integral test the serier converges too!

for the second one just look at the example in your book it should give you step by step on what all of that means taylor polynomials and remainders are tedious and your book will show more steps than I would

This post was edited by brigadier on May 2 2014 01:48am

Weren't you doing Algebra 2 ~4-5 months ago?

Congratulations!

Wow you guys are awesome seriously.